Compatible Poisson brackets of hydrodynamic type

Abstract

Some general properties of compatible Poisson brackets of hydrodynamic type are discussed, in particular: (1) an invariant differential-geometric criterion of the compatibility based on the Nijenhuis tensor; (2) the Lax pair with a spectral parameter governing compatible Poisson brackets in the diagonalizable case; (3) the connection of this problem with the class of surfaces in Euclidean space which possess nontrivial deformations preserving the Weingarten operator.

14 pages

References in corpus (1)

• On Integrability of the Equations for Nonsingular Pairs of Compatible Flat Metrics

Cited by in corpus (36)

• Deformations of Semisimple Bihamiltonian Structures of Hydrodynamic Type
• Nijenhuis Geometry
• On Hamiltonian perturbations of hyperbolic systems of conservation laws
• A simple way of making a Hamiltonian system into a bi-Hamiltonian one
• Applications of Nijenhuis geometry II: maximal pencils of multihamiltonian structures of hydrodynamic type
• Exact Poisson pencils, $\tau$-structures and topological hierarchies
• Hamiltonian operators of Dubrovin-Novikov type in 2D
• Bihamiltonian Cohomologies and Integrable Hierarchies II: the Tau Structures
• Compatible metrics on a manifold and non-local bi-Hamiltonian structures
• Applications of Nijenhuis geometry III: Frobenius pencils and compatible non-homogeneous Poisson structures
• Applications of Nijenhuis geometry: Nondegenerate singular points of Poisson-Nijenhuis structures
• Deformations of non semisimple Poisson pencils of hydrodynamic type
• Generalized symmetries, conservation laws and Hamiltonian structures of an isothermal no-slip drift flux model
• Central invariants revisited
• Classification of bi-Hamiltonian pairs extended by isometries
• Reciprocal transformations and flat metrics on Hurwitz spaces
• Variational Bihamiltonian Cohomologies and Integrable Hierarchies II: Virasoro symmetries
• Compatible Dubrovin-Novikov Hamiltonian operators, Lie derivative and integrable systems of hydrodynamic type
• Weakly Nonlocal Hamiltonian Structures: Lie Derivative and Compatibility
• Variational Bihamiltonian Cohomologies and Integrable Hierarchies I: Foundations
• Jacobi Structures of Evolutionary Partial Differential Equations
• Miura-reciprocal transformations and localizable Poisson pencils
• Tri-Hamiltonian Structure of the Ablowitz-Ladik Hierarchy
• Poisson pencils: reduction, exactness, and invariants
• New Hamiltonian formalism and Lagrangian representations for integrable hydrodynamic type systems
• Liouville canonical form for compatible nonlocal Poisson brackets of hydrodynamic type, and integrable hierarchies
• Quasi-Frobenius algebras and their integrable N-parametric deformations generated by compatible (N x N)-metrics of constant Riemannian curvature
• Inherited structures in deformations of Poisson pencils
• Darboux coordinates for Hamiltonian structures defined by Novikov algebras
• On the compatible weakly-nonlocal Poisson brackets of Hydrodynamic Type
• Hamiltonian Structures and Reciprocal Transformations for the $r$-KdV-CH Hierarchy
• On surfaces with prescribed shape operator
• On compatible metrics and diagonalizability of non-locally bi-Hamiltonian systems of hydrodynamic type
• Lax pairs for the equations describing compatible nonlocal Poisson brackets of hydrodynamic type, and integrable reductions of the Lame equations
• Compatible metrics of constant Riemannian curvature: local geometry, nonlinear equations and integrability
• Description of compatible differential-geometric Poisson brackets of the first order